Construct complete compound-interest recurrence models

A deposit is made before any interest is credited. If n counts completed compounding periods, what does V(0) represent?

Keep the nominal annual rate at 6%. Set the number of equal periods to 1, 4 and 12. Observe how the period rate and its multiplier change. Some other slider positions display rounded approximations

At 12 periods per year, which multiplier gives one monthly update?

Give both equations, not just a multiplier

For an initial balance of $4,000 with 5% interest compounded yearly, the first balance is 4,000 and each annual update multiplies the current balance by 1.05. Use V and n in both equations

1. 1Include the starting equation The principal is the balance at index 0, before interest is added \(\displaystyle V_0=P\)
2. 2State the complete recurrence Both equations and the meaning of n belong in a written financial model. The automatic questions supply the time unit and check both equations as one response \(\displaystyle \begin{gathered}V_0=P\\V_{n+1}=RV_n\end{gathered}\)

Can this multiplier-only model describe a loan while fixed repayments are being made each month?

Nominal is not effective

The divide-by-m rule applies to a stated nominal annual rate convertible m times per year. An effective annual rate already describes the full-year change and cannot simply be divided by m. Every annual-to-period question here explicitly supplies a nominal rate

1. 1Match the rate to one index step For a nominal annual percentage r and m equal compounding periods per year, divide r by m. The index counts those periods, not automatically years \(\displaystyle i=\frac{r}{m}\%\)
2. 2Convert percentage to multiplier The new balance retains the current balance and adds that period’s interest \(\displaystyle R=1+\frac{r}{100m}\)

Change what the index counts

At a nominal annual rate of 7.2% compounded monthly, one step is a month. The monthly percentage is 0.6%, not 7.2%. For an initial balance of $12,000, the following model has n counting months

A savings account earns 0.4% per month, compounded monthly. What rate should be used for one monthly update?

Use the stated year convention

These ideal questions use 1, 4, 12, 26, 52 or 365 periods per year for yearly, quarterly, monthly, fortnightly, weekly or daily compounding respectively. Real financial products may use other day-count conventions; use the convention stated in the problem. For nominal 3.65% with 365 daily periods, n counts days

Two positive balances at indices 0 and 1 are supplied. Is that alone enough to prove every later balance follows the same compound-interest multiplier?

Recover a model from exact balance data

Suppose a stated constant compound-interest model has exact balances $8,000 initially and $8,064 one month later. The multiplier is later balance divided by earlier balance. These rows are exact, not rounded observations

1. 1Interpret the rate The monthly interest rate is 0.8%. Its nominal annual equivalent under monthly compounding is 9.6%, not its effective annual rate

A saver deposits an additional $50 after interest is credited each month. Which statement is correct?

Write your own investment model before continuing

Ungraded written task. Choose a positive starting investment and a plausible fixed nominal annual rate. Choose monthly or quarterly compounding. Define the balance variable and the meaning of n, state no-cash-flow/no-fee assumptions, write both recurrence equations and use them to calculate the balances after the first and second compounding periods. Explain why your multiplier matches one period. Record your own work before viewing one possible model on the next screen

One possible investment model

Example only: invest $6,000 at nominal 4.8% p.a., compounded monthly, with no deposits, withdrawals or fees. Let V(n) be dollars after n months, with the initial deposit at n=0. Monthly interest is 0.4%, so the period multiplier is 1.004. The ideal second balance is 6,048.096 dollars; report $6,048.10 only if a cents answer is requested. Compare the same features in your independently chosen model

Write your own loan model before continuing

Ungraded written task. Choose a positive loan balance and a plausible fixed nominal annual rate compounded quarterly. Define n, state that no repayments, extra borrowing or fees occur, and construct both recurrence equations. Calculate the first quarter’s balance and interest separately. Explain how allowing repayments would change the model. Record your own work before viewing an example

One possible loan model

Example only: an initial outstanding balance of $15,000 at nominal 8% p.a., compounded quarterly. With no repayments, further borrowing or fees, quarterly interest is 2% and n counts quarters. The first balance is $15,300, but the interest added is $300. A repayment would need an additional subtraction at the specified time, not a claim that the same pure multiplier still fits

What is checked, and what needs written review

The automatic full-rule questions check both recurrence equations jointly. The index unit and no-cash-flow assumptions are supplied. The separate conversion question checks two linked numerical entries, not two unrelated successes. Your chosen-context tasks require written explanation and self/teacher review; neither recognising an assumption nor entering a multiplier replaces that explanation

1. 1State the financial assumptions A constant multiplier alone assumes a constant rate with no deposits, repayments, withdrawals, extra borrowing or fees. Keep unrounded balances in the ideal model; round only a requested reported amount